Quiver varieties and Weyl group actions. (English) Zbl 0958.20036

Lusztig, George

Ann. Inst. Fourier 50, No. 2, 461-489 (2000).

For a finite graph of type \(ADE\) with set of vertices \(I\), H. Nakajima associated to \(\mathbf{v,w}\in\mathbb{N}^I\) a “quiver variety” \(\mathcal M(\mathbf{v,w})\) and constructed a Weyl group action on the cohomology of \(\sqcup_{\mathbf v}\mathcal M(\mathbf{v,w})\) using techniques of hyper-Kähler geometry [see Duke Math. J. 76, No. 2, 365-416 (1994; Zbl 0826.17026), Duke J. Math. 91, No. 3, 515-560 (1998)]. In the paper under review it is given an alternative construction of this Weyl group action, based not on hyper-Kähler geometry, but on techniques of intersection cohomology. This gives in fact a refinement of the Weyl group action.

Reviewer: Vladimir L.Popov (Wien)

MSC:

20G10	Cohomology theory for linear algebraic groups
20G05	Representation theory for linear algebraic groups
14L30	Group actions on varieties or schemes (quotients)

Keywords:

quiver varieties; Weyl groups; intersection cohomology; Weyl group actions

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